## Operations Research - May 2014

### Mechanical Engineering (Semester 7)

TOTAL MARKS: 80

TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.

(2) Attempt any **three** from the remaining questions.

(3) Assume data if required.

(4) Figures to the right indicate full marks.

### Write short note on the following. (Any four)

**1 (a)** Feasible solution in L.P model(5 marks)
**1 (b)** Objectives of Operations Research(5 marks)
**1 (c)** Assumptions in sequencing model.(5 marks)
**1 (d)** Degeneracy in the LP problem(5 marks)
**1 (e)** Game theory(5 marks)
**1 (f)** Difference between assignment and transportation models(5 marks)
**2 (a)** Solve the LP problem by using Simplex method.

Maximize Z=2x_{1}+5x_{2}

Subjected to X_{1}+4X_{2}≤24

3X_{1}+X_{2}≤21

X_{1}≤ 9

X_{1 } , X_{2}≥ 0(10 marks)
**2 (b)** Solve the following assignment problem.

I | II | III | IV | V | |

1 | 11 | 17 | 8 | 16 | 20 |

2 | 9 | 7 | 12 | 6 | 15 |

3 | 13 | 16 | 15 | 12 | 16 |

4 | 21 | 24 | 17 | 28 | 26 |

5 | 14 | 10 | 12 | 11 | 13 |

**3 (a)**A machine operator has to perform two operations turning and threading on a number of different jobs. The time required to perform these operations (in minutes) for each job is known. Determine the order in which the jobs should be processed in order to minimize the tow! time required to turn out all the jobs.

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JOB | Time for turning (min) | Time for threading (min) |

1 | 3 | 8 |

2 | 12 | 10 |

3 | 5 | 9 |

4 | 2 | 6 |

5 | 9 | 3 |

6 | 11 | 1 |

Also determine the total processing time and idle times for turning and threading operations.(10 marks)

**3 (b)**ABC manufacturing company wishes to develop a monthly production schedule for the next three months. Depending upon the sales commitments, the company can either keep the production constant. allowing the fluctuations in inventory, or inventories can be maintained at a constant level. with fluctuating production. Fluctuating production necessitates in working overtime, the cost of which is estimated to be double the normal production cost of Rs.12 per unit. Fluctuating inventories result in inventory carrying cost of Rs. 2 per unit per month. If the company fails to fulfil its sales commitment, it incurs a shortage cost of Rs 4 per unit month. The production capacities for the next three months are shown below:

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Production capacity | |||

Month | Regular | Overtime | Sales |

1 | 50 | 30 | 60 |

2 | 50 | 0 | 120 |

3 | 60 | 50 | 40 |

Determine the optimal production schedule(10 marks)

**4 (a)**Food X contains 6 units of vitamin A per gram and 7 units of vitamin B per gram and costs 12 paise per gram. Food Y contains 8 unit of vitamin A per gram and 12 units of vitamin B per gram. The daily minimum requirement of vitamin A and vitamin B is 100 units and 120 units respectively. Find the minimum cost of product mix.(10 marks)

**4 (b)**A chemical company produces two products, X and Y. Each unit of product X requires 3 hours on operation I and 4 hours on operation II, while each unit of product Y requires 4 hours on operation I and 5 hours on operation II. Total available time for operations I and II is 20 hours and 26 hours respectively. The production of each unit of product Y also results in two units of a by-product Z at no extra cost.

Product X sells at profit of Rs. 10/unit. while Y sells at profit of Rs 20/unit. By-product 2 brings a unit profit of Rs 6 if sold; in case if it is not sold, the destruction cost is Rs 4/unit. Forecasts indicate that not more than 5 units of Z can be sold. Formulate the L. P. model to determine the quantities of X and Y to be produced. keeping Z in mind. So that the profit earned is maximum.(10 marks)

**5 (a)**Solve the following L.P.problem by dynamic programming approach.

Maximize Z=3X

_{1}+5X

_{2>/sub> Subjected to X1≤ 4 X2 ≤ 6 3X2+2X2 ≤ 18 X1, X2 ≥ 0}(10 marks)

**5 (b)**Explain in detail the method of solving the L P problem by Two phase method(10 marks)

**6 (a)**Reduce the following game by dominance and find the game value

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PLAYER A | PLAYER B | |||

I | II | III | IV | |

I | 3 | 2 | 4 | 0 |

II | 3 | 4 | 2 | 4 |

III | 4 | 2 | 4 | 0 |

IV | 0 | 4 | 0 | 8 |

**6 (b)**Discuss types of simulation models. What are advantages and limitations of simulation models?(10 marks)

**7 (a)**The cost of a machine is Rs. 6,100 and its scrap value is Rs. 100. The maintenance costs found from experience are as follows:

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Year | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

Maintenance Cost | 100 | 250 | 400 | 600 | 900 | 1200 | 1600 | 2000 |

When should the machine be replaced?(10 marks)

**7 (b)**The annual demand of a product is 10,000 units. Each unit costs Rs. 100 if orders placed in quantities below 200 units but for orders of 200 and above the price is Rs 95. The annual inventory holding costs is 10 per cent of the value of the item and ordering cost is Rs 5 per order. Find the economic lot size.(10 marks)